52. Corollary. Two fixed points, or one point and a certain direction, determine the position of a straight line.

53. If a straight line were turned upon two of its points as fixed pivots, no part of the line would change place. So any figure may revolve about a straight line, while the position of the line remains unchanged.

This property is peculiar to the straight line. If the curve BC were to revolve upon

the two points B and C as pivots, then the straight line con

B

C

necting these points would remain at rest, and the curve would revolve about it.

A straight line about which any thing revolves, is called its AXIS.

54. Axiom of Distance.-The straight line is the shortest which can join two points.

Therefore, the distance from one point to another is reckoned along a straight line.

55. There have now been given two postulates and two axioms. The science of geometry rests upon these four simple truths.

The possibility of every figure defined, and the truth of every problem, depend upon the postulates.

Upon the postulates, with the axioms, is built the demonstration of every principle.

SURFACES.

56. Surfaces, like lines, are classified according to their uniformity or change of direction.

A PLANE is a surface which never varies in direction. A CURVED SURFACE is one in which there is a change. of direction at every point.

THE PLANE.

57. The plane surface and the straight line have the same essential character, sameness of direction. The plane is straight in every direction that it has.

A straight line and a plane, unless the extent be specified, are always understood to be of indefinite

extent.

58. Theorem. A straight line which has two points in a plane, lies wholly in it, so far as they both extend.

For if the line and surface could separate, one or the other would change direction, which by their definitions. is impossible.

59. Theorem.-Two planes having three points common, and not in the same straight line, coincide so far as they both extend.

Let A, B, and C be three points which are not in one straight line, and let these points A be common to two planes, which may be designated by the letters m and

p. Let a straight line

pass through the points A and

E

B, a second through B and C, and a third through A and C.

Each of these lines (58) lies wholly in each of the planes m and p. Now it is to be proved that any point D, in the plane m, must also be in the plane p.

Let a line extend from D to some point of the line. AC, as E. The points D and E being in the plane m, the whole line DE must be in that plane; and, therefore, if produced across the inclosed surface ABC, it will meet one of the other lines AB, BC, which also lie in that plane, say, at the point F. But the points F and E

Geom.-3

are both in the plane p. Therefore, the whole line FD, including the point D, is in the plane p.

In the same manner, it may be shown that any point which is in one plane, is also in the other, and therefore the two planes coincide.

60. Corollary.-Three points not in a straight line, or a straight line and a point out of it, fix the position of a plane.

61. Corollary.—That part of a plane on one side of any straight line in it, may revolve about the line till it meets the other part, when the two will coincide (53).

EXERCISES.

62. When a mechanic wishes to know whether a line is straight, he may apply another line to it, and observe if they coincide.

In order to try if a surface is plane, he applies a straight rule to it in many directions, observing whether the two touch throughout.

The mason, in order to obtain a plain surface to his marble, applies another surface to it, and the two are ground together until all unevenness is smoothed away, and the two touch throughout.

What geometrical principle is used in

each of these operations?

In a diagram two letters suffice to mark

a straight line. Why?

But it may require three letters to designate a curve. Why?

DIVISION OF SUBJECT.

63. By combinations of lines upon a plane, PLANE FIGURES are formed, which may or may not inclose an

area.

By combinations of lines and surfaces, figures are

formed in space, which may or may not inclose a vol

ume.

In an elementary work, only a few of the infinite variety of geometrical figures that exist, are mentioned, and only the leading principles concerning those few.

Elementary Geometry is divided into PLANE GEOMETRY, which treats of plane figures, and GEOMETRY IN SPACE, which treats of figures whose points are not all in one plane.

In Plane Geometry, we will first consider lines without reference to area, and afterward inclosed figures. In Geometry in Space, we will first consider lines and surfaces which do not inclose a space; and afterward, the properties of certain solids.

PLANE GEOMETRY.

CHAPTER III.

STRAIGHT LINES.

64. Problem.-Straight lines may be added together, and one straight line may be subtracted from another.

For a straight line may be produced to any extent. Therefore, the length of a straight line may be increased by the length of another line, or two lines may be added together, or we may find the sum of several lines (35).

One

Again, any straight line may be applied to another, and the two will coincide to their mutual extent. line may be subtracted from another, by applying the less to the greater and noting the difference.

65. Problem.-A straight line may be multiplied by any number.

For several equal lines may be added together. 66. Problem.-A straight line may be divided by another.

By repeating the process of subtraction.

67. Problem.-A straight line may be decreased in any ratio, or it may be divided into several equal parts.

This is a corollary of the postulate of extent (35).